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Bias 14: Conjunction Fallacy

We work in an international import/export firm with an extremely large distribution of our products worldwide. We have just received a new stock of toys and we discover that one of them is a nice doll with a lovely pink dress who says ”Ti amo” when we press a button on her back.

Who do we think is the most probable buyer of this doll?

A boy

An Italian mother

An Italian

We are tempted to answer that the most probable buyer is an Italian mother, however, the real answer is counter-intuitive. Let’s have a look at the probabilities of these events, assuming that the company is implanted in equally in each country.

Conjunction fallacy:

The set of ”Italian mother” is included in the set of ”Italian” so that we have necessarily P(”Italian mother”)<P(”Italian”)

Neglect of prior probability:

P(”Italian”)=60.000.000/ 6.000.000.000=1/100
P(“Boy”)=1/2

According to Bayesian rules, we have:

P(“Boy”l”Buyer”)~P(”Boy”)*P(”Buyer”l”Boy”) and P(“Italian”l”Buyer”)~(P(”Italian”)*P(”Buyer”l”Italian”)
P(“Boy”l”Buyer”)~1/2*P(”Buyer”l”Boy”) and P(“Italian”l”Buyer”)~1/100*P(”Buyer”l”Italian”).

If we assume the probability that potential buyers are Italian is higher than the potential buyers being boys, that means we assume that the probability of an Italian buying this doll is 50 times higher than a boy buying this doll, which is a lot!

More generally speaking, we tend to forget the prior probability when we intuitively solve Bayesian problems.